where we have used that \(\displaystyle \frac{\ \mathrm {d}\phi _i}{\ \mathrm {d}t} = 0\)  and \(\displaystyle \frac{\ \mathrm {d}{x}}{\ \mathrm {d}t} = \bf 0\)  per our quasistatic model.
We have assumed that the boundary terms vanish for simplicity, but they can easily be accounted for following, e.g., [1]}, [2]}.
